Problem: Simplify; express your answer in exponential form. Assume $x\neq 0, p\neq 0$. $\dfrac{{(x^{5}p^{-4})^{4}}}{{(x^{-3}p^{5})^{-5}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(x^{5}p^{-4})^{4} = (x^{5})^{4}(p^{-4})^{4}}$ On the left, we have ${x^{5}}$ to the exponent ${4}$ . Now ${5 \times 4 = 20}$ , so ${(x^{5})^{4} = x^{20}}$ Apply the ideas above to simplify the equation. $\dfrac{{(x^{5}p^{-4})^{4}}}{{(x^{-3}p^{5})^{-5}}} = \dfrac{{x^{20}p^{-16}}}{{x^{15}p^{-25}}}$ Break up the equation by variable and simplify. $\dfrac{{x^{20}p^{-16}}}{{x^{15}p^{-25}}} = \dfrac{{x^{20}}}{{x^{15}}} \cdot \dfrac{{p^{-16}}}{{p^{-25}}} = x^{{20} - {15}} \cdot p^{{-16} - {(-25)}} = x^{5}p^{9}$